The Unsolved Lollipop Problem - Numberphile

[00:00] What's a lollipop? Tell me about

[00:01] lollipops. Yeah, well, this is a

[00:03] mathematical lollipop. What it consists

[00:05] of

[00:06] it is a circle and a

[00:08] on a stick.

[00:10] And if you continued the stick, it would

[00:12] pass through the center of the circle.

[00:14] This part of the stick is invisible and

[00:16] the stick is infinite. So, it's a a

[00:18] circle with a perpendicular stick coming

[00:21] out of it, perpendicular to the edge of

[00:22] the circle. That's a lollipop. And we've

[00:25] got one of them and it looks like that

[00:26] and it divides up the region. If it's If

[00:29] we're doing them in the sand, it divides

[00:31] up the beach into the part that's inside

[00:34] the circle and the rest of the beach.

[00:36] And so, one lollipop divides the beach,

[00:40] the paper, into two regions, inside the

[00:43] circle and all the rest. Because this

[00:46] goes off to infinity. What if we have

[00:48] two lollipops? This is actually a little

[00:50] tricky. If you have two lollipops, how

[00:53] many regions can you make by overlapping

[00:56] them? Well, you could do this. You could

[00:58] put your other lollipop here and its

[01:02] stick could go here. And then, if we did

[01:04] that, how many regions would we get?

[01:06] One, two, three, four, five, six. We get

[01:11] We could easily get six regions.

[01:13] >> not get seven? Does that If these go

[01:15] forever, is the other side of the thing

[01:17] not a seventh? It's seven.

[01:18] >> It is seven, right.

[01:19] >> Yeah, yeah, you're right. Yeah, there

[01:20] you go. I know my stuff. You know your

[01:22] stuff.

[01:22] >> Seven. All right. So, each stick goes

[01:24] off to infinity. So, with with two, we

[01:27] can certainly get seven. And that's

[01:29] okay, but it's not great. You can

[01:31] actually get 10 pieces if you do it

[01:33] right. And what you do is you make sure

[01:36] that the circles of the

[01:38] lollipops overlap in just a little

[01:41] sliver. And then, you make sure the

[01:43] stick of the first lollipop cuts the

[01:47] circle of the second lollipop. Okay,

[01:50] yeah. Yeah, and then, scalpel, blue,

[01:54] we get

[01:56] this. It's going to go through the the

[01:58] center imaginary through the center of

[02:01] that. It's going to cut across this gulf

[02:04] where the two circles

[02:07] and then it's going to cut that one. So,

[02:09] I'll give it a little bit of space.

[02:13] Looks like that. And what you get let's

[02:16] count the pieces. How many pieces do we

[02:18] get when we do that? It's very careful.

[02:20] We all right. We have first of all, we

[02:22] have below the the sticks and above the

[02:25] sticks. They they two infinite regions.

[02:27] All right, one, two, then three and

[02:30] four. Yeah. And then five for the

[02:32] sliver, six and seven down here.

[02:35] >> Yeah.

[02:36] And then eight, nine, 10. So, we get 10

[02:38] pieces. You can show using higher

[02:40] mathematics Euler's formula that the

[02:43] crucial thing

[02:45] if you want to get the most pieces

[02:48] what you need to focus on are the

[02:50] intersections, the crossings between one

[02:54] lollipop and the other lollipop. And the

[02:56] crossings it's it's crucial that you

[02:59] notice. You using Euler's formula you

[03:01] can show that the important thing is to

[03:04] get the most intersections between the

[03:07] lines of one lollipop and the lines of

[03:09] the other lollipop. And the

[03:11] intersections

[03:12] the there are three kinds of

[03:14] intersections. There are intersections

[03:16] where the circle part cross and you can

[03:19] see I've managed to make this circle and

[03:21] this circle cross in two points.

[03:24] There's also

[03:25] intersections where the stick of one

[03:28] lollipop crosses

[03:30] the circle of the other lollipop and

[03:33] we'd like each of them to happen twice.

[03:35] So, we can see that blue stick crosses

[03:38] the other lollipop twice. Correct. And

[03:41] vice versa, this the stick crosses this

[03:44] one here and here. And then the sticks

[03:46] themselves can cross.

[03:48] And they do here at just once. Sticks

[03:51] either cross or they don't. So, with two

[03:53] two circles, you get seven

[03:54] intersections.

[03:56] And there's a formula, the number of

[03:58] pieces equals intersections number of

[04:03] intersections plus n, the number of

[04:06] sticks plus one.

[04:07] With two, n equals two,

[04:10] we got seven intersections. I showed you

[04:13] 2 + 2 + 2 + 1, seven, and then two

[04:17] because we got two circles plus one, and

[04:19] that's 10. And that's the best we can do

[04:21] with two lollipops. And if we had three

[04:23] lollipops, ideally, we'd make each pair

[04:27] of lollipops intersect in this way. And

[04:30] it looks like this. All right. It is

[04:33] very tricky to draw. Okay, so the new

[04:35] lollipop is the green one. That's

[04:37] >> one at the bottom. Okay.

[04:39] >> Yeah. And it intersects the red one in

[04:42] the same way that the blue and the red

[04:43] intersected, and it intersects the blue

[04:45] one in the same way. Each pair of

[04:48] lollipops here

[04:50] meet in seven intersections. And the

[04:53] stick from the third new lollipop Yeah,

[04:56] it's going up. But it's slightly

[04:58] off-center, so that it doesn't it

[05:00] doesn't intersect with the intersection

[05:02] of the first two sticks. Correct.

[05:04] >> To maximize our sections.

[05:05] >> Yeah, you never want to have three

[05:06] things meeting at a point because you

[05:08] make a tiny little change and you pick

[05:11] up one piece, one region. So, that So,

[05:13] that stick is slightly off Slightly

[05:16] off-center, and it's true. You might

[05:18] think I'm fudging this, but actually, if

[05:21] you work to multiple precision and you

[05:24] draw it carefully, I did actually draw

[05:26] it carefully.

[05:27] And you can see it in the OEIS entry for

[05:30] this sequence. So, the So, how many

[05:32] pieces did we end up with for three

[05:34] lollipops? We need to know how many

[05:36] intersections there are. Each pair

[05:38] intersect in seven points. So, there are

[05:40] seven intersections there, seven there,

[05:43] and seven there. And none of them have

[05:45] been counted twice.

[05:46] >> No, they're all distinct. You can see

[05:49] check I'm very careful not to have any

[05:51] triple points or higher. So, we got 21

[05:55] intersections. All right, n equals

[05:57] three. All right. Three lollipops. The

[06:00] number of intersections is equal to 7 +

[06:02] 7 + 7 each because each we got three

[06:06] lollipops and each pair meets in seven

[06:08] points.

[06:09] >> Yeah. And they're all different. So,

[06:10] that's 21 and then the formula is this.

[06:15] We add n, which is three, and we add

[06:17] one, we get 25. So, with three

[06:20] lollipops, we get 25 regions. But, where

[06:23] are we going to put the fourth lollipop?

[06:25] >> That's all I can think of.

[06:25] >> This is

[06:26] really, really hard. Yeah.

[06:29] Cuz you cuz if you put it up You put it

[06:31] up there, it's not going to meet

[06:33] >> No. No.

[06:34] >> Yeah. No, it is really hard. Where does

[06:37] the fourth lollipop go?

[06:38] >> Where does the And I tried and I did

[06:41] various drawings. And we know what we

[06:43] want. We want the maximum number of of

[06:45] intersections between all the pairs of

[06:48] lollipops. With with four lollipops,

[06:51] we've got six intersections. So,

[06:53] ideally, we'd get 6 * 7 42

[06:57] intersections.

[06:59] Let's give people some thinking time.

[07:04] Yeah.

[07:04] >> All right. What's the answer? Well, it

[07:07] what didn't come very easily. On

[07:10] Christmas Eve, I posted a message to the

[07:12] Secret Santa mailing list explaining

[07:14] this problem and asking for help. It

[07:17] said it for I said with four lollipops,

[07:20] it's really tricky. And at 1 minute past

[07:23] midnight, I got an email from a couple

[07:26] of old friends who said that they could

[07:29] get

[07:30] 43 regions. The maximum would be 47.

[07:34] If you could get every pair to meet in

[07:37] seven points, you'd get 42 + 4 + 1,

[07:42] you'd get 47 regions. They got close,

[07:44] but

[07:45] but not not very close. And how And how

[07:49] did they uh place their circles to do

[07:50] that? Well, they took my drawing of

[07:54] three circles, and they added a fourth

[07:56] circle, which they got by perturbing one

[07:59] of the three a little bit. So, we still

[08:02] crossed most of the things the same way.

[08:04] >> They didn't perturb one of the existing

[08:06] circles. They They added

[08:08] >> They They put their new lollipop on top

[08:11] of the existing lollipop and perturbed

[08:13] that one. Yes. They took a copy of the

[08:15] red one and perturbed it a bit. Maybe

[08:17] they changed the the the diameter a

[08:19] little bit, I'm not sure. And they

[08:21] changed the angle of the stick. And that

[08:23] gave them 43. And that gave them 43

[08:26] regions.

[08:27] Cool.

[08:29] It was pretty good. Yeah. And for the

[08:31] first 12 hours, that was the world

[08:33] record. And then

[08:35] 2 minutes past noon on Christmas Day, I

[08:39] got an email from

[08:41] someone on the Sequence Fans mailing

[08:43] list who I've never met, although since

[08:45] we've talked on Zoom.

[08:49] He was able to get 44 regions. But

[08:54] later, he got it up to 45. 45 regions.

[08:57] And what's more, he proved that was

[08:59] optimal. So, there was no point in

[09:02] anyone trying to get more. You could

[09:04] theoretically have gotten 46 or 47, but

[09:08] you can't. He proved that 45 is best

[09:10] possible. Yeah. What he did was really

[09:12] extraordinary.

[09:15] He took those three, and he magnified He

[09:17] modified them a little bit. He made one

[09:20] rather bigger than the other two, about

[09:22] twice as big. And then he blew blew it

[09:25] up, magnified it by a factor of 100. So,

[09:28] these circles got really, really huge.

[09:32] And when you looked at the edge of the

[09:33] circle, it was the circle was so huge,

[09:36] it the edge looked almost like a

[09:38] straight line.

[09:39] And I will show you what those straight

[09:41] lines looked like.

[09:43] And here's a picture of how it looks

[09:46] after he's blown it up. That green line

[09:48] is part of a gigantic circle. Yeah, and

[09:51] that's the stick of the green lollipop.

[09:52] >> That's the stick of the green line.

[09:54] And the red [clears throat]

[09:55] >> And the red is also that's

[09:57] >> The red is the

[09:59] It's this

[10:00] red circle magnified so that that arc

[10:04] looks like a straight line. And there it

[10:06] is. And that's the stick. And then blue,

[10:09] this is the blue circle. Yeah. And

[10:12] here's the blue stick. So he he

[10:14] magnified it. And then he very cleverly

[10:17] put a fourth circle on top of the place

[10:20] where the sticks come together. So that

[10:22] little black lollipop, that's the new

[10:25] lollipop and it's miniaturized right in

[10:28] the mess between the other three.

[10:29] Exactly. Yes, brilliant. So that So

[10:33] there it is. And if we go back and look

[10:36] at the previous picture,

[10:38] the extra fourth lollipop, the black

[10:41] lollipop is actually here. You just

[10:43] can't see it. It's so tiny. It's in

[10:46] blots of ink where the three circles

[10:49] and the three sticks come together.

[10:52] Neil, you were telling me before that

[10:55] 47

[10:56] was the fantasy. Yes.

[10:58] The best that's possible is 45. In fact,

[11:02] yes.

[11:02] >> Yes. Where did we lose? Where did we we

[11:05] lose the two? What's the problem here?

[11:06] The problem is really that it's putting

[11:09] down the fourth stick so it crosses all

[11:12] the other circles in the right way. Ah,

[11:15] because this black stick obviously looks

[11:17] like it goes out this way towards the It

[11:19] crosses It doesn't It crosses the red

[11:22] stick here.

[11:23] And it crosses the blue stick here. So

[11:25] the

[11:26] the black stick is okay, but it's also

[11:29] got to cross all the circles. And the

[11:31] circles have to cross all the circles.

[11:33] Oh, cuz it doesn't cross the green

[11:35] circle, does it? Ever. Obviously.

[11:36] >> No, obviously. The black stick will

[11:38] never cross the green circle. It will

[11:39] cross the green stick.

[11:41] >> Yeah.

[11:42] Eventually, a few miles away. Not the

[11:45] green circle.

[11:46] >> But not the green circle. Okay. So,

[11:48] we've lost two two goals. We're down by

[11:50] two goals and that's the best you can

[11:52] do.

[12:04] Where's the fifth lollipop going to go?

[12:06] We have

[12:08] estimates

[12:09] for that. And again, it's by taking one

[12:13] of the four and making a copy of it and

[12:17] perturbing it a bit.

[12:19] Okay. Shaking it a bit and putting it

[12:21] down. And in fact, Jonas worked out how

[12:25] to do that with taking

[12:28] copies of all four of these and putting

[12:31] them down and jiggling them a little

[12:33] bit. We have bounds with with five

[12:35] circles. All we know, I mean, we know a

[12:38] lot. It's either 71 or 72 regions.

[12:42] But we don't know which of those two.

[12:44] >> We don't know which of those two. Are

[12:46] people working on this? Uh

[12:48] I don't know. They might after they've

[12:50] seen this video. I hope they will. We We

[12:53] think the answer's probably 71.

[12:55] But that's just a guess.

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